0

I always found it a curiousity that in the symmetry groups of the known fundamental forces we find the nice arithmetic progression $1,2,3$: first there is $\DeclareMathOperator{\U}{U}\DeclareMathOperator{\SU}{SU}\U(\color{red} 1)$, then $\SU(\color{red}2)$, and finally $\SU(\color{red}3)$.

Questions:

  1. Is this merely a coincidence?
  2. Were there reasons to believe, that after having found $\U(1)$ and $\SU(2)$ symmetries, that the next symmetry to look out for is $\SU(3)$?
  3. Have people considered $\SU(4)$ symmetries, or is there a reason it stops after $3$?

I have heard of $\SU(5)$ theories, but those seem motivated not by the arithmetic progression, but by the fact that $\SU(5)$ contains $\U(1)\times\SU(2)\times\SU(3)$.

Qmechanic
  • 201,751
M. Winter
  • 523
  • 1
    To reopen this post (v2) consider to only ask 1 question per post. The 4th question is a duplicate of https://physics.stackexchange.com/q/119190/2451 Related: https://physics.stackexchange.com/q/116831/2451 – Qmechanic Aug 30 '23 at 14:31
  • @Qmechanic I removed the second part of the question. I could also pack the remaining three subquestions into a single sentence but what would be the use of that. I hope you agree it is more readable like this. – M. Winter Aug 30 '23 at 14:32
  • 1
    The On-Line Encyclopedia of Integer Sequences has 30035 results for series that start with "1,2,3". – Mauricio Aug 30 '23 at 14:36
  • @Mauricio Admittedly, the sequence could continue differently after $1,2,3,...$. But I do not ask to extrapolate from this data. I ask whether physicists have an idea at all for why these numbers and therefore how it might continue based on that. – M. Winter Aug 30 '23 at 14:40
  • You are plain uninformed. There have been several speculative theories in both particle and nuclear physics utilizing SU(4), SU(6), you name it, depending on fundamental pieces (fermions) considered and the links amongst them. Staring at these groups and expecting to numerological home in on the "music of the spheres" might well be a mug's game, but who knows? Nobody. Linked. – Cosmas Zachos Aug 30 '23 at 17:01
  • @CosmasZachos I take away from your comment that there is no theoretical reason to stop at $3$. I also kind of read between the lines that the groups that we find (and that we find only one of each) seems to be a crude fact of nature, and we were lucky that they are so simple and that we could identify them so quickly. If so, this answers 1 and 3, leaving 2 (but gives me the feeling that the answer is No). – M. Winter Aug 30 '23 at 17:30
  • The reason for 2 is happenstance. – Cosmas Zachos Aug 30 '23 at 17:33

0 Answers0